7/26/2018 SECOND HOURLY PRACTICE V Maths 21a, O.Knill, Summer 2018

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1 7/26/28 SECOND HOURLY PRACTICE V Maths 2a, O.Knill, Summer 28 Name: Start by printing your name in the above box. Try to answer each question on the same page as the question is asked. If needed, use the back or the next empty page for work. If you need additional paper, write your name on it. Do not detach pages from this exam packet or unstaple the packet. Provide details to all computations except for problems -3. Please write neatly. Answers which are illegible for the grader can not be given credit. No notes, books, calculators, computers, or other electronic aids can be allowed. You have 9 minutes time to complete your work Total:

2 Problem ) True/False questions (2 points). No justifications needed. Mark for each of the 2 questions the correct letter. No justifications are needed.) T F 2) T F If (,) is not a critical point of f(x,y), then the level curve through (,) intersects a sufficiently small circle x 2 +y 2 = r 2 in exactly two points. 3) T F A point (x,y ) at which the gradient is zero, is called a discriminant. 4) T F 5) T F 6) T F The relation f xx = f yy holds everywhere as a consequence of Clairot s theorem. The tangent plane to any point of the surface (x + y + z) 2 = 4 is either x+y +z = 2 or x+y +z = 2. Fubini s theorem implies that for any function f(x,y) of two variables, we have f(x,y) dxdy = 4 3 f(x,y) dxdy. 7) T F Saddle points with positive discriminant are called monkey saddles. 8) T F 9) T F 2 If f(x,y) = and R is a region in the plane, then Rf(x,y) dxdy is the area of the region. There is a function f(x,y) for which the linearization L(x,y) at a point is the function itself. ) T F The directional derivative at a local maximum is negative in every direction. ) T F 2) T F If r(t) is a curve on the sphere g(x,y,z) = x 2 +y 2 +z 2 = then g( r(t)) r (t) =. If not at a critical point, the length f(,) of the gradient is equal to the directional derivative into the direction of the gradient. 3) T F If D > and f(,) = and f xx < then f yy <. The tion 4) T F 5) T F 6) T F 7) T F x f(x,y) dydx = y f(x,y) dydx. The surface area of the sphere of radius L was written in class as 2π L 2 sin(φ) dφ. If f(x,y) depends on one variable only, then the discriminant D satisfies D = at every critical point. The gradient vector f(x,y ) is a vector in the plane perpendicular to the level curve of f(x,y) through (x,y ). 8) T F We have seen that e x2 dx = π. 9) T F 2) T F In three dimensions, the gradient of f has always the form f(x,y) = [f x,f y,] T If f xx (x,y) = f yy (x,y) everywhere, then every critical point is a saddle point. 2

3 Problem 2) ( points) No justifications are needed a) (6 points) Match the regions with the integrals. Each integral matches one region A F. A B C D E F Enter A-F Integral y x x f(x,y) dxdy f(x,y) dydx x f(x,y) dydx 2 y y f(x,y) dxdy 2 y 2 f(x,y) dxdy 2 f(x,y) dydx x b)(4 points) Name the partial differential equations correctly. Each equation matches one name. Fill in -4 Name Wave Transport Burgers Heat Equation Number PDE u x u y = 2 u xx u yy = 3 u x u yy = 4 u y +uu x u xx = Problem 3) ( points) (No justifications are needed.) 3

4 This summer, Oliver studies zeta functions of networks, functions of two variables which contain information about the network. They encode for example the frequencies at which the network resonates, if it is excited. Zeta functions are defined for geometric objects like surfaces. For a circle, it is the Riemann zeta function f(x,y) for which finding the roots is a prize of million dollars written out. The contour map below belongs to the zeta function of the network seen to the right. Check each box which is true. Don t split hairs. If something is not true, it should be clearly false. The point (,) is a local maximum. The point (2,4) is a local minimum. Close to (2, 6), there is a saddle point. The partial derivative f x (,) is positive. Under the constraint x =, f has a maximum near y =. Under the constraint y = 8, f has has a minimum near x =. The gradient at (, 8) is longer than the one at (2,8). The global maximum of f is at the boundary. Near (, ), we have D [,] Tf =. Near (, 3), we have D (,)/ 2 f =. 4

5 Problem 4) ( points) The material to build a traffic light is g(x,y) = 6+6πxy +3πx 2 = 2 is fixed (the radius of each cylinder is x and the height is y and the constant 6 is the material for the back plate). We want to build a light for which the shaded region with volume f(x,y) = 3πx 2 y is maximal. Use the Lagrange method. Problem 5) ( points) a) (8 points) Find all the critical points of the function and classify them. f(x,y) = xy x x 2 y +x 2 b) (2 points) Are there global maxima and minima of f(x,y)? Problem 6) ( points) 5

6 A decorative paper lantern is made of 8 surfaces. Each is parametrized by r(t,z) = [zcos(t),zsin(t),z] T with t 2π and z and then translated or rotated. Find the total surface area of the lantern. Problem 7) ( points) It is the year 23, and you have been appointed royal math magician to prince charming just born to Kate, the Duchess of Cambridge and prince William. You show the teenage prince how to estimate ( )/3 by finding a linearization of the function f(x,y) = (x/y) /3 = x /3 y /3 at the point (x,y ) = (25,27). The young prince for a moment even ignores his iphone 7 implanted in his skull and murmurs thats soooh coool! Problem 8) ( points) 6

7 a) (6 points) Find the tangent plane to the surface x 2 xyz +y 2 +z 2 = 2 at the point (,,). The surface makes a nice connection piece between two cylinders. b) (4 points) Find the tangent line to the curve x /3 y /3 = at (,). Problem 9) ( points) You find yourself in the desert at the point A = (a,), completely dehydrated and almost dead. You want to reach the point B = (b,) as fast as possible but you can not reach it without water. There is an lake inside the ellipsoid g(x,y) = x 2 +2y 2 =. The amount of effort you need to go from a point (x,y) to a point (u,v) is assumed to be (x u) 2 + (y v) 2 (this is justified by the fact that if you walk for a long time, you walk less andlessefficientlysothatwalkingtwiceaslongwilltake you 4 times as much effort). Find the path of least effort which connects A with X = (x,y) and then with B. a) Which function f(x,y) do you extremize? The parameters a,b are constants. b) Write down the Lagrange equations. c) Solve the Lagrange equations in the case a =,b =. Problem ) ( points) 7

8 a) (5 points) Integrate f(x,y) = x 2 y 2 over the unit disk {x 2 +y 2 }. b) (5 points) An evil integral! θ 2 r 2 drdθ. 8